Download Parameter-free testing of the shape of a probability distribution

BioSystems 90 (2007) 509–515
Parameter-free testing of the shape of a probability distribution
M. Brooma , P. Nouvelletb , J.P. Baconb , D. Waxmanb,∗
a
School of Science and Technology, University of Sussex, Falmer, Brighton BN1 9QG, Sussex, UK
b School of Life Sciences, University of Sussex, Falmer, Brighton BN1 9QG, Sussex, UK
Received 30 August 2006; received in revised form 5 December 2006; accepted 6 December 2006
Abstract
The Kolmogorov–Smirnov test determines the consistency of empirical data with a particular probability distribution. Often,
parameters in the distribution are unknown, and have to be estimated from the data. In this case, the Kolmogorov–Smirnov test
depends on the form of the particular probability distribution under consideration, even when the estimated parameter-values are
used within the distribution. In the present work, we address a less specific problem: to determine the consistency of data with a
given functional form of a probability distribution (for example the normal distribution), without enquiring into values of unknown
parameters in the distribution. For a wide class of distributions, we present a direct method for determining whether empirical
data are consistent with a given functional form of the probability distribution. This utilizes a transformation of the data. If the
data are from the class of distributions considered here, the transformation leads to an empirical distribution with no unknown
parameters, and hence is susceptible to a standard Kolmogorov–Smirnov test. We give some general analytical results for some of
the distributions from the class of distributions considered here. The significance level and power of the tests introduced in this work
are estimated from simulations. Some biological applications of the method are given.
© 2006 Elsevier Ireland Ltd. All rights reserved.
Keywords: Kolmogorov–Smirnov test; Non-parametric method; Functional form of a distribution
1. Introduction
A standard method of testing whether measured data
are consistent with a particular continuous probability
distribution is the Kolmogorov–Smirnov (KS) test
(see standard statistical texts, for example Hogg and
Tanis, 2006). The essence of the test is to compare the
maximum distance, termed DKS , between the empirical
cumulative distribution and the particular cumulative
distribution of interest. In many practical cases, the
distribution of interest contains one or more parameters
that are not known a priori, but have to be estimated
from the data. When this applies, the distance statistic,
DKS , that underlies the KS test, no longer has a universal
∗
Corresponding author. Tel.: +44 1 273 678 559;
fax: +44 1 273 678 937.
E-mail address: [email protected] (D. Waxman).
distribution, i.e. one that is independent of the distribution under consideration. The procedure introduced by
Lilliefors (1969) to test an empirical distribution, when
parameters have to be determined from the data, is to
carry out numerical simulations and thereby produce
the distribution of the test statistic that is relevant for
the particular distribution under consideration.
Here we consider a less specific question than directly
testing whether measured data are consistent with a
particular distribution. Our aim is to determine whether
the measured data are consistent with the general
functional form (i.e. shape) of a cumulative distribution
of interest, independent of the value of any unknown
parameters upon which the distribution depends. Thus,
we might wish to test whether our data could consistently be interpreted as arising from an exponential
distribution, without having any interest in the value of
the single parameter characterising the exponential dis-
0303-2647/$ – see front matter © 2006 Elsevier Ireland Ltd. All rights reserved.
doi:10.1016/j.biosystems.2006.12.002
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M. Broom et al. / BioSystems 90 (2007) 509–515
Table 1
Lists some of the distributions arising from random variables X of
the form X = a + bξ, where a is the location parameter, b the scale
parameter and ξ is the standard random variable
Distribution
Location
parameter
Scale
parameter
Standard random
variable ξ
1
2
3
4
5
α
0
0
μ
α
β−α
λ
λ
σ
β
U
− log(U)
± log(U)
Z
eZ
For a wide range of important probability distributions (including those
mentioned in this paper), the functional forms are given in texts on
probability theory or mathematical statistics (such as Hogg and Tanis,
2006 or Weiss, 2006). Some of the distributions in this table have
standard names and symbols: (1) is the uniform distribution and written
as U[α, β]; (2) is the exponential distribution and written as exp(λ);
(3) with the sign, ± picked at random with equal probability, we have
a reflected exponential distribution; (4) is a normal distribution and
written as N(μ, σ 2 ); (5) corresponds to a particular case of a lognormal
distribution that has been horizontally translated by an amount α. We
shall simply refer to it as a lognormal distribution in the present work.
tribution. Focussing solely on the functional form of the
distribution means our test is non-parametric in nature.
The approach adopted here allows a direct test of the
data when the random variables underlying the distribution of interest may be expressed in the form:
X = a + bξ
(1)
where a is the location parameter, b the non-zero scale
parameter and ξ is the standard random variable (i.e.
a random variable from a known distribution, with
no unknown parameters). Before we proceed with
the derivation of our results, we note that there are a
number of well-known examples of random variables
that may be defined by an expression of the form of Eq.
(1). In terms of U ≡ U[0, 1], a random number that is
uniformly distributed on [0, 1], and Z, a standard normal
random variable (i.e. with mean zero and variance
unity), some examples of random variables of the form
in Eq. (1) are given in Table 1.
2. Methodology
We shall consider the general problem shortly, but first
investigate the special case where the location parameter
of the distribution, a, is zero.
2.1. Method 1: location parameter is zero
The simplest case arises when the location parameter
a has the value a = 0. This is the case where the particular distribution under consideration is specified by a
single parameter, b. The shape of the distribution of the
random variable X, of Eq. (1), is determined from the
distribution of the standard random variable ξ. Denoting
the probability density of ξ by fξ (x) and its cumulative
distribution by Fξ (x), the probability density and cumulative distribution of X ≡ bξ are fξ (x/b)/b and Fξ (x/b),
respectively. These evidently depend upon the parameter
b. We can derive a distribution related to that of X that is
independent of the parameter b as follows.
Let Xi and Xj represent independent draws (i = j) of
X = bξ from its distribution and let us use the notation
|X|< = min(|Xi |, |Xj |) and |X|> = max(|Xi |, |Xj |).
We show in Appendix A that the cumulative distribution and probability density of:
R0 =
|X|<
|X|>
(2)
are, for r in the range 1 ≥ r ≥ 0, given by:
FR0 (r) = 2
−∞
fR0 (r) = 2
∞
∞
−∞
fξ (y)[Fξ (r|y|) − Fξ (−r|y|)] dy
(3)
|y|fξ (y)[fξ (r|y|) + fξ (−r|y|)] dy
(4)
Outside the range 1 ≥ r ≥ 0, fR0 (r) vanishes, hence
FR0 (r) vanishes for r < 0 and is unity for r > 1. Because the parameter b cancels between the numerator
and denominator in Eq. (2), it follows that R0 is independent of any parameters and hence FR0 (r) is determined
solely by the parameter-free cumulative distribution of
the standard random variable ξ, namely Fξ (x). A standard KS test may then be directly applied, since FR0 (r)
is a known distribution with no unknown parameters.
Thus, we can proceed by testing whether the empirical
distribution of the data is consistent with FR0 (r) without
having to first determine any parameters from the data.
In practical applications, there remains the question
of how to construct the set of values of the R0 statistic from the set of realised X values, i.e. from the set
{x1 , x2 , . . . , xN }, where N is the number of measured
values. Assuming N is even (or is made so, by discarding the final x), we use each x value only once, by forming
N/2 ratios, for example {x1 /x2 , x3 /x4 , . . . , xN−1 /xN }.
These are converted to R0 values by (i) taking their absolute value and then (ii) taking the smaller of the resulting
absolute value or its reciprocal. For example, the corresponding R0 value obtained from x1 and x2 is the smaller
of |x1 /x2 | and |x2 /x1 |.
In the discussion, we give the rationale for the particular form of the R0 statistic adopted.
M. Broom et al. / BioSystems 90 (2007) 509–515
2.2. Method 2: general case
It is straightforward to observe that in the general case,
where a = 0, the quantity (Xi − Xj )/(Xk − Xl ) equals
(ξi − ξj )/(ξk − ξl ) and hence is independent of a and b.
Accordingly we define:
R=
|X − X|<
|X − X|>
(5)
where |X − X|< = min(|Xi − Xj |, |Xk − Xl |) and
|X − X|> = max(|Xi − Xj |, |Xk − Xl |) for i, j, k and l
all different.
Following the same logic as in Method 1 (see
Appendix A), the cumulative distribution and probability
density of R are, for 1 ≥ r ≥ 0, given by:
∞
FR (r) = 2
fξ−ξ (y)[Fξ−ξ (r|y|) − Fξ−ξ (−r|y|)] dy
−∞
(6)
fR (r) = 2
∞
−∞
|y|fξ−ξ (y)[fξ−ξ (r|y|) + fξ−ξ (−r|y|)] dy
(7)
where fξ−ξ (•) is the probability density of ξi − ξj ,
∞
i.e. fξ−ξ (x) = −∞ fξ (y)fξ (y − x) dy, and Fξ−ξ (•) is
the corresponding cumulative distribution. Outside the
range 1 ≥ r ≥ 0, fR (r) vanishes, hence FR (r) vanishes
for r < 0 and is unity for r > 1.
By construction, the cumulative distribution of R, i.e.
FR (r), and its probability density, fR (r), are independent
of unknown parameters present in the probability density
of X, and hence as in Method 1, a standard KS test may
be directly employed.
In this more general case, we again use each X
data value only once in the construction of the set of
R values and the simplest way to proceed, assuming
N is divisible by 4 (or is made so by discarding the
necessary final few x’s) is to form a set of N/4 values of R given by {(x1 − x2 )/(x3 − x4 ), (x5 − x6 )/(x7 −
x8 ), . . . , (xN−3 − xN−2 )/(xN−1 − xN )}. We then, again,
take the absolute value of these and select the smaller of
the resulting absolute value or its reciprocal.
3. Application to some standard distributions
Details of the calculations are given in Appendix A.
We note that by definition of R0 or R, these random
variables lie in the range [0, 1] and, as a result, all probability densities are non-zero only for r in the range [0, 1].
The corresponding cumulative distributions are zero for
r < 0 and unity for r > 1. The results given in this sec-
511
tion will only apply for the range of r where the cumulative distributions exhibit non-trivial behaviour, namely
1 ≥ r ≥ 0, without this restriction being further stated in
the results.
3.1. Exponential distribution
First consider the case where a = 0 and ξ ∼ exp(1),
so fξ (x) = exp(−x) for x ≥ 0 and vanishes otherwise.
Method 1 (above) leads, via Eq. (3), to the cumulative
distribution of R0 being given by FR0 (r) = 2r/(1 + r)
and to the probability density fR0 (r) = 2/(1 + r)2 .
Method 2 (above), for a = 0, yields, via Eq. (6), to
identical results to those of Method 1, i.e. to FR (r) =
2r/(1 + r) and fR (r) = 2/(1 + r)2 .
3.2. Reflected exponential distribution
For the reflected exponential distribution, fξ (x) =
exp(−|x|)/2, Method 1 leads, via Eq. (3), to the cumulative distribution of R0 being given by FR0 (r) = 2r/(1 +
r) and to the probability density fR0 (r) = 2/(1 + r)2 .
Method 2 leads, via Eq. (6), to a cumulative distribution of R being given by FR (r) = (r/2)(3 + 9r +
4r2 )/(1 + r)3 and to the probability density fR (r) =
(3/2)(1 + 4r + r 2 )/(1 + r)4 .
3.3. Normal distribution
√
We have ξ ∼ N(0, 1) so fξ (x) = exp(−x2 /2)/ 2π.
Method 1 leads, via Eq. (3), to the cumulative distribution
of R0 being given by FR0 (r) = (4/π) arctan(r) and to
the probability density fR0 (r) = 4/[π(1 + r 2 )]. Method
2, which applies when a = 0, leads to identical results to
those of Method 1: FR (r) = (4/π) arctan(r) and fR (r) =
4/[π(1 + r2 )].
3.4. Uniform distribution
Consider first the case with a = 0. Using Method 1,
the cumulative distribution of R0 is given by FR0 (r) = r
(i.e. coinciding with that of a uniform distribution on
[0, 1]) and to the probability density fR0 (r) = 1. For the
case where a = 0 we use Method 2, and obtain a cumulative distribution of R given by FR (r) = (r/3)(4 − r) and
to the probability density fR (r) = (2/3)(2 − r).
4. Numerical tests
In this section, we give results from simulated data
from various distributions, and compare the simulated
data to the distributions FR0 (•) or FR (•) evaluated
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M. Broom et al. / BioSystems 90 (2007) 509–515
above. We performed two sets of comparisons, first using
Method 1 and then Method 2.
4.1. Method
To evaluate the characteristics of the statistical tests
proposed here, we investigated the associated errors. We
note that it is usual to distinguish two kinds of error:
(i) A type I error corresponds to rejecting a valid null
hypothesis. The probability of a type I error is termed
the significance level of the test.
(ii) A type II error corresponds to accepting an invalid
null hypothesis. The complement of the probability
of a type II error gives the power of the test.
A good statistical test should have low probabilities
of producing both type I and type II errors.
Generally, to perform a KS test, one estimates the
maximum distance of the cumulative distribution derived from observation, Fobs (•), from the theoretical one,
such as FR (•). This distance, denoted DKS , is known
as the Kolmogorov–Smirnov statistic, and is defined as
DKS = maxx |Fobs (x) − FR (x)|. Using DKS , it is possible to assess if an observed distribution is effectively
indistinguishable from a theoretical distribution (this is
the null hypothesis). Thus, if the distance between the
distributions, DKS , is greater than the critical distance
associated with the KS test, for the sample size under
consideration, then the null hypothesis is rejected. For
our purposes we use a p value of 0.05. We thus reject
the null hypothesis when a value of DKS is obtained that
would arise, by chance, in ≤ 5% of the cases where the
null hypothesis is valid.
One of the major constraints of the KS test, as explained above, is that any parameters appearing in the
theoretical distribution must be known. The change in
variable adopted here (from X to R0 or R; see Eqs. (2)
and (5)), allows us, in a number of cases of interest, to
test if a random variable follows a theoretical distribution of given functional form (i.e. shape), independent
of any parameters in the distribution of X.
To find the significance level of the test (the probability of type I errors), we generated random numbers from
exponential, normal and uniform distributions. These
were used in Methods 1 and 2 (for the variables R0 and
R described above). We evaluated the distance, DKS , between the cumulative distribution of a simulated sample
and the particular distribution used to generate the sample. The null hypothesis was listed as rejected at the 5%
level when DKS was greater than the critical value of this
statistic for the sample size adopted. Critical values of
DKS follow from a standard KS test and can be found in
published tables. Any such rejections correspond to the
false conclusion that the sample comes from a distribution different from the specified theoretical distribution,
and indicate a type I error of the test.
We next generated random numbers from lognormal
and reflected exponential distributions. These distributions are, respectively, very similar in shape to exponential and normal distributions. The data from these
simulations were tested to see if:
(i) The sample drawn from a lognormal distribution is
effectively indistinguishable from a theoretical exponential distribution.
(ii) The sample drawn from a reflected exponential distribution is effectively indistinguishable from a theoretical normal distribution.
By evaluating DKS we estimated the probability of (i)
or (ii) being accepted as an indication of type II errors
i.e. as an indication of the power of the test.
For each comparison, we computed DKS for different sample sizes, N, ranging from N = 10 to 2000. For
each sample size, v replicate samples were generated
and v was chosen so that v × N = 3 × 106 was the total
number of random numbers generated. The DKS values
obtained were used to estimate the probability of acceptance of the null hypothesis (type II errors), for each
comparison of distributions, and for each sample size.
Our estimates of the various errors are summarised in
Table 2.
5. Results
5.1. The significance level of the test
We obtained an estimate of the significance level of
the test from our data by taking a large number of samples
from a specified distribution and finding the proportion
where the null hypothesis was erroneously rejected. As
we were using the standard KS test with a significance
level of 5% we would naturally expect an estimate in the
region of 5% in each case. We did this for the exponential,
normal and uniform distributions and as expected, we
did indeed, find that approximately 5% of the sample
distributions led to rejection of the null hypothesis.
5.2. Power of the test
We investigated the power of the test by looking at
the proportion of cases where the null hypothesis was
incorrectly accepted. This was carried out for sample
M. Broom et al. / BioSystems 90 (2007) 509–515
513
Table 2
A summary of type I and type II errors is given for various distributions
Sample size
8
20
48
100
200
500
752
1000
2000
Estimate of the probability of type I error
Uniform (e) vs.
Exponential (e)
uniform (t)
vs. exponential (t)
Normal (e) vs.
normal (t)
Estimate of the probability of type II error
Lognormal (e) vs.
Reflected exponential
exponential (t)
(e) vs. normal (t)
Method 1
Method 2
Method 1
Method 2
Method 1
Method 2
Method 1
Method 2
Method 1
Method 2
0.050
0.051
0.051
0.050
0.047
0.049
0.051
0.057
0.049
0.050
0.049
0.049
0.051
0.048
0.046
0.050
0.042
0.046
0.050
0.050
0.050
0.050
0.051
0.047
0.050
0.050
0.050
0.050
0.049
0.053
0.051
0.052
0.047
0.052
0.047
0.057
0.050
0.050
0.049
0.050
0.052
0.050
0.045
0.051
0.051
0.050
0.050
0.050
0.050
0.051
0.044
0.050
0.049
0.044
0.952
0.937
0.902
0.824
0.650
0.207
0.044
0.008
0.000
0.943
0.942
0.933
0.920
0.892
0.804
0.740
0.666
0.396
0.929
0.905
0.842
0.730
0.523
0.150
0.041
0.013
0.000
0.947
0.944
0.937
0.927
0.900
0.832
0.769
0.713
0.467
The null hypothesis is that “the observed distribution is drawn from the theoretical distribution”. The label e or t following the name of a distribution
in this table serves to distinguish between a distribution that is of effectively empirical (e) or theoretical (t) origin. Thus, X values are drawn from
the distributions labelled e, and are our simulation of experimental data. The theoretical distribution of the variables R0 or R (Eqs. (2) and (5)) are
determined from the distribution labelled t and some examples of these distributions are given in Section 3. Under Method 1, all of the distributions
labelled e, that were used to generate simulated X data, had a location parameter of a = 0 and a scale parameter of b = 3. The simulated X data
were then transformed to R0 data, according to Eq. (2) and a standard KS test was then employed (as described in the main text). Under Method
2, all distributions labelled e had a location parameter of a = 2 and a scale parameter of b = 3. The simulated X data were then transformed to
R data, according to Eq. (5) and a standard KS test was, again, then employed. We used the tables of Neave (1989) for the critical values of the
Kolmogorov–Smirnov statistic.
distributions that were similar in shape to the theoretical distribution that was used for the null hypothesis.
We thus tested (i) a lognormal sample distribution (distribution 5 in Table 1) against a theoretical exponential
distribution (distribution 2 in Table 1) and (ii) a reflected
exponential sample distribution (distribution 3 in Table
1) against a theoretical normal distribution (distribution
4 in Table 1). We found that the test of Method 1 is, for a
given sample size, sometimes less powerful than that of a
standard KS test and sometimes more powerful. Method
2 performed less well than the test of Method 1. Since
differences of two random variables are typically closer
to being normally distributed than a single random variable, this may provide partial explanation of the relative
performances of Methods 1 and 2, rather than attributing it solely to the reduction of the size of the data set of
Method 1 (size N/2), to that of Method 2 (size N/4).
6. Biological applications of the method
The method presented in this paper can have a wide
range of applications in biology and other scientific
fields. Here we give as examples some possible behavioural ecology applications.
The method of parameter-free testing of the shape of
a probability distribution was motivated from an analysis of the foraging behaviour of ants. In an experiment
(Nouvellet et al., in preparation), times were recorded
at which individual ants left their nest to explore a new
area. The ants that left the nest, and arrived in the new
area, were not able to return or communicate in any way
with their nest mates. The distribution of time intervals
that developed over the course of the experiment between each successive ant leaving the nest was calculated. There is the strong possibility that the distribution
of time intervals will depend on the actual duration of
the experiment. This follows since ants cannot, during
the course of the experiment, return to the nest; hence the
number within the nest declines over time and the rate at
which ants leave the nest may also decline. Thus any distribution, for example, that characterising time intervals,
may change over time; in other words any parameters it
depends upon may be time dependent. We can however
test if the functional form (or shape) of the distribution
is time independent, if we make the assumption that during a typical time interval between two ants leaving the
nest, any time-dependent parameters within the distribution change negligibly. Given this, we used Method 1
of this work to test whether the time intervals between
ants leaving the nest had the functional form of an exponential distribution, with all time dependence carried
by the single parameter characterising the exponential
distribution. Thus, Method 1 (as fully described above)
allows the use of the Kolmogorov–Smirnov test on the
data to test whether the time intervals are exponentially
distributed, without knowledge of the parameter in the
distribution. Full details of the experiment and related
experiments along with a detailed analysis of the data
will be presented elsewhere (Nouvellet et al., in preparation).
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M. Broom et al. / BioSystems 90 (2007) 509–515
This specific analysis can be extended, for example,
to any type of social behaviour involving the movement
of numbers of individuals. Examples are bees, wasps or
termites leaving the nest or arriving at a food or water
source. Additionally the variation of parameter(s) can be
subsequently inferred to understand the manner in which
it varies. Hence we can decouple different phenomena—
the purely statistical from the values of parameters and
their variation, using the methods of this paper.
It is clear that any behaviour that approximately repeats over time, and which might be influenced by external factors, could be analysed in this way to test whether
there is a distribution of constant shape underlying the
phenomenon. External factors could be the size of the
‘source population’ in the nest, the quantity or quality
of the ‘sink food source’, but also environmental factors
such as day-length.
On this last topic, we note that many temporal behaviours (hunting, singing, diving, onset of activity) appear to show a constantly shaped distribution that is
shifted by the timing of an external event. Birds sing
at sunrise (Fisler, 1962), but each day sunrise occurs at a
different time, thus preventing the direct pooling of data
taken at different times of year. Similarly, hunting behaviour (Van Orsdol, 1984), basking behaviour (Ciofolo
and Boissier, 1992), onset of activity (Aschoff, 1966;
Semenov et al., 2000) often occurs at certain times that
exhibit variation over the course of a year.
There is thus a wide range of behavioural studies that
could utilize the statistical approach described in this
paper, where the functional form of a distribution is important and needs to be established.
7. Discussion
In this paper, we have introduced a method of data
transformation which allows us to test whether data come
from a particular functional form of a distribution, irrespective of the values of unknown parameters in the distribution. The distribution is of the form of Eq. (1) with
ξ following a fully specified distribution. Many distributions can be written in the form of Eq. (1), and such a test
is consequently of interest for a wide variety of practical
situations.
For the transformed data, as given in Eqs. (2) and
(5), we have specified the precise form of the distribution for several important standard distributions. Thus for
these distributions, the method may be directly applied
without further calculation. We have also given general
expressions for the form of the probability density and
cumulative distribution of the transformed data, in terms
of the density and distribution of the underlying random
variable, ξ, so that such expressions can be derived for
other cases.
We have examined the power of our test in a variety
of situations to examine the usefulness of the method.
There does not appear to be a consistent advantage or
disadvantage of Method 1, compared with the original
Kolmogorov–Smirnov test for comparable data, where
the distribution is known. For example, if in a standard KS test, we wish to discriminate between (i) data
which are lognormally distributed, i.e. distribution 5 of
Table 1 with α = 0 and β = 1, with (ii) an exponential distribution (distribution 2 of Table 1, with λ = 1)
we then find that it requires a sample size of N 100
so the different distributions can be discriminated in
approximately 95% of all cases. By contrast, Method
1 of the present work requires N 750 to discriminate the functional form of the two distributions to the
same level of power. However, applying the same procedure to (i) the reflected exponential distribution (distribution 3 of Table 1, with λ = 1) and (ii) a normal
distribution (distribution 4 of Table 1, with μ = 0 and
σ = 1), a standard KS test requires a sample size of
N 850 to discriminate the distributions in approximately 95% of all cases. By contrast, Method 1 of the
present work requires N 700 to discriminate the functional form of the two distributions to the same level of
power.
The power from the more general Method 2 is noticeably less than Method 1 (see Table 2) and thus there is
a corresponding need for large data sets, which means
that its range of applicability will be restricted.
In Section 2.1 of this work, we adopted a particular
function of Xi /Xj , namely that of Eq. (2), as the statistic at the heart of Method 1, and a related quantity for
Method 2. The virtue of the choice made is that R0 lies in
a compact range, whereas e.g. Xi /Xj can cover a very
wide range of values, caused by potentially small denominators. We have found significantly different powers of a test based on R0 or Xi /Xj ; the wider range of
the test statistic being accompanied by a significantly
lower power to discriminate between different distributions.
The original motivation of this work arose from
analysing the foraging behaviour of ants, as described
in the previous section. We envisage that there may be a
substantial range of other applications of the statistical
methods presented here.
Acknowledgement
It is a pleasure to thank John Welch for helpful discussions.
M. Broom et al. / BioSystems 90 (2007) 509–515
Appendix A
In this appendix, we provide some details of the calculations underlying results in the main text.
First we find the general form for the cumulative distribution in Method 1. We can write
FR0 (r) = Prob(|ξ1 |/|ξ2 | < r||ξ1 | < |ξ2 |) = Prob(|ξ1 | <
r|ξ2 |||ξ1 | < |ξ2 |). For r > 1 we have FR0 (r) = 1 and
we shall henceforth restrict analysis to the range
1 ≥ r ≥ 0, where FR0 (r) exhibits nontrivial behaviour. We then have FR0 (r) = 2Prob(|ξ1 | < r|ξ2 |) =
2Prob(r|ξ2 | > ξ1 > −r|ξ2 |). It quickly follows that
∞
FR0 (r) = 2 −∞ fξ (y)[Fξ (r|y|) − Fξ (−r|y|)] dy.
To
obtain the probability density, we differentiate the above
with respect to r and the result of the main text, Eq. (4),
follows immediately.
To find the cumulative distribution and the probability
density for Method 2, we simply note that if ξ1 , ξ2 , ξ3 and
ξ4 are independently and identically distributed (i.i.d.),
then ζ1 = ξ1 − ξ2 and ζ2 = ξ3 − ξ4 are also i.i.d. with
zero mean, thus meeting the conditions of Method 1.
Hence FR (r) = Prob(|ζ1 |/|ζ2 | < r||ζ1 | < |ζ2 |) and the
above results apply with the corresponding density and
cumulative distribution of ξi − ξj . In this way we obtain
Eqs. (6) and (7) of the main text.
Next we calculate the distributions of R0 and R for
some specific cases of interest. Because of the definitions
of R0 and R (Eqs. (2) and (5)), the cumulative distribution
and probability density of these random variables are
only non-zero for r in the range 1 ≥ r ≥ 0 and for brevity,
we shall only give the form of the distributions in this
range of r.
515
A.4. Reflected exponential distribution, Method 2
We have fξ (y) = e−|y| /2 and find fξ−ξ (y) = (1 +
|y|)e−|y| /4. Application of Eqs. (6) and (7) lead to
FR (r) = (r/2)(3 + 9r + 4r 2 )/(1 + r)3 and to fR (r) =
(3/2)(1 + 4r + r 2 )/(1 + r)4 .
A.5. Normal distribution, Method 1
√
2
We have fξ (y) = e−y /2 / 2π and application of
Eqs. (3) and (4) lead to FR0 (r) = (4/π) arctan(r) and
fR0 (r) = 4/[π(1 + r 2 )].
A.6. Normal distribution, Method 2
√
2
We find that fξ−ξ (y) = e−y /4 / 4π and application
of Eqs. (6) and (7) yield identical results, to those found
for Method 1, for the cumulative distribution and probability density of R0 for a normal distribution.
A.7. Uniform distribution, Method 1
Direct application of Eq. (3) for ξ ∼ U(0, 1) quickly
yields FR0 (r) = r and hence fR0 (r) = 1 (corresponding
to R0 ∼ U(0, 1)).
A.8. Uniform distribution, Method 2
For ξ ∼ U(0, 1) we find fξ−ξ (y) = (1 − |y|) for 1 >
y > −1 and is zero elsewhere. Direct application of Eq.
(6) yields FR (r) = (r/3)(4 − r) and fR (r) = (2/3)(2 −
r).
A.1. Exponential distribution, Method 1
References
We calculate FR0 (r) by direct application of Eq. (3).
We have fξ (y) = e−y for y ≥ 0 and zero otherwise. We
also have Fξ (y) = (1 − e−y ) for y ≥ 0 and zero otherwise. It follows that FR0 (r) = 2r/(1 + r) and by differentiation or from Eq. (4) we obtain fR0 (r) = 2/(1 + r)2 .
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A.2. Exponential distribution, Method 2
We have fξ (y) = e−y for y ≥ 0 and zero otherwise.
We find fξ−ξ (y) = e−|y| /2 and application of Eqs. (6)
and (7) leads to FR (r) = 2r/(1 + r) and to fR (r) =
2/(1 + r)2 .
A.3. Reflected exponential distribution, Method 1
We have fξ (y) = e−|y| /2 and results for this distribution coincide with the results for the exponential distribution, for Method 2.